# CompNO: A Novel Foundation Model approach for solving Partial Differential Equations

arXiv: https://arxiv.org/abs/2601.07384 · HTML: https://mlsanity.com/paper/2601.07384/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 24 of 24)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs](https://mlsanity.com/paper/2403.12553/) (2024)
- [PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics](https://mlsanity.com/paper/2409.09811/) (2024)
- [UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation](https://mlsanity.com/paper/2403.07187/) (2024)
- [VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction](https://mlsanity.com/paper/2411.16063/) (2024)
- [PhysiX: A Foundation Model for Physics Simulations](https://mlsanity.com/paper/2506.17774/) (2025)
- [Guiding continuous operator learning through Physics-based boundary constraints](https://mlsanity.com/paper/2212.07477/) (2022)
- [BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamics](https://mlsanity.com/paper/2501.18972/) (2025)
- [Defining Foundation Models for Computational Science: A Call for Clarity and Rigor](https://mlsanity.com/paper/2505.22904/) (2025)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution](https://mlsanity.com/paper/2402.10238/) (2024)
- [Fine-Tune Language Models as Multi-Modal Differential Equation Solvers](https://mlsanity.com/paper/2308.05061/) (2023)

## Built on by (most cited first, 5 of 5)

- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [SCNO: Spiking Compositional Neural Operator -- Towards a Neuromorphic Foundation Model for Nuclear PDE Solving](https://mlsanity.com/paper/2604.11625/) (2026)
- [Compositional Neural Operators for Multi-Dimensional Fluid Dynamics](https://mlsanity.com/paper/2605.11691/) (2026)
- [Compositional Embedding Architecture for Physical Field Prediction in Componentized Aerospace Systems](https://mlsanity.com/paper/2610.00237/) (2026)
- [COSMOS: Soft Mechanism Mixtures with Verifiable Routing for Long-Horizon PDE Forecasting](https://mlsanity.com/paper/2610.04427/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent Scientific Foundation Models (SFMs) aim to alleviate this cost by learning universal surrogates from large collections of simulated systems, yet they typically rely on monolithic architectures with limited interpretability and high pretraining expense. In this work we introduce Compositional Neural Operators (CompNO), a compositional neural operator framework for parametric PDEs. Instead of pretraining a single large model on heterogeneous data, CompNO first learns a library of Foundation Blocks, where each block is a parametric Fourier neural operator specialized to a fundamental differential operator (e.g. convection, diffusion, nonlinear convection). These blocks are then assembled, via lightweight Adaptation Blocks, into task-specific solvers that approximate the temporal evolution operator for target PDEs. A dedicated boundary-condition operator further enforces Dirichlet constraints exactly at inference time. We validate CompNO on one-dimensional convection, diffusion, convection--diffusion and Burgers' equations from the PDEBench suite. The proposed framework achieves lower relative L2 error than strong baselines (PFNO, PDEFormer and in-context learning based models) on linear parametric systems, while remaining competitive on nonlinear Burgers' flows. The model maintains exact boundary satisfaction with zero loss at domain boundaries, and exhibits robust generalization across a broad range of Peclet and Reynolds numbers. These results demonstrate that compositional neural operators provide a scalable and physically interpretable pathway towards foundation models for PDEs.